The restoration lemma is a classic result by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [PODC '01], which relates the structure of shortest paths in a graph $G$ before and after some edges in the graph fail. Their work shows that, after one edge failure, any replacement shortest path avoiding this failing edge can be partitioned into two pre-failure shortest paths. More generally, this implies an additive tradeoff between fault tolerance and subpath count: for any $f, k$, we can partition any $f$-edge-failure replacement shortest path into $k+1$ subpaths which are each an $(f-k)$-edge-failure replacement shortest path. This generalized result has found applications in routing, graph algorithms, fault tolerant network design, and more. Our main result improves this to a multiplicative tradeoff between fault tolerance and subpath count. We show that for all $f, k$, any $f$-edge-failure replacement path can be partitioned into $O(k)$ subpaths that are each an $(f/k)$-edge-failure replacement path. We also show an asymptotically matching lower bound. In particular, our results imply that the original restoration lemma is exactly tight in the case $k=1$, but can be significantly improved for larger $k$. We also show an extension of this result to weighted input graphs, and we give efficient algorithms that compute path decompositions satisfying our improved restoration lemmas.

翻译：恢复引理是Afek、Bremler-Barr、Kaplan、Cohen和Merritt [PODC '01] 提出的一个经典结果，它关联了图 $G$ 中某些边发生故障前后最短路径的结构。他们的研究表明，在一条边故障后，任何避免该故障边的替换最短路径可以被划分为两条故障前的最短路径。更一般地，这意味着容错性与子路径数量之间存在加性权衡：对于任意 $f, k$，我们可以将任何 $f$ 边故障替换最短路径划分为 $k+1$ 条子路径，每条子路径都是一个 $(f-k)$ 边故障替换最短路径。这一推广结果已应用于路由、图算法、容错网络设计等领域。我们的主要改进将此结果提升为容错性与子路径数量之间的乘性权衡。我们证明，对于所有 $f, k$，任何 $f$ 边故障替换路径可以被划分为 $O(k)$ 条子路径，每条子路径都是一个 $(f/k)$ 边故障替换路径。我们还给出了一个渐近匹配的下界。特别地，我们的结果表明，原始恢复引理在 $k=1$ 情形下是精确紧的，但对于更大的 $k$ 可以显著改进。我们还将此结果推广到带权输入图，并设计了满足改进恢复引理的路径分解的高效算法。